Average Error: 30.7 → 0.3
Time: 14.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x}
double f(double x) {
        double r618749 = 1.0;
        double r618750 = x;
        double r618751 = cos(r618750);
        double r618752 = r618749 - r618751;
        double r618753 = r618750 * r618750;
        double r618754 = r618752 / r618753;
        return r618754;
}
↓
double f(double x) {
        double r618755 = x;
        double r618756 = sin(r618755);
        double r618757 = r618756 / r618755;
        double r618758 = r618757 * r618757;
        double r618759 = 1.0;
        double r618760 = cos(r618755);
        double r618761 = r618759 + r618760;
        double r618762 = r618758 / r618761;
        return r618762;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

    \[\frac{1 - \cos x}{x \cdot x}\]
  2. Using strategy rm

  3. Applied flip--30.8

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

  4. Simplified15.4

    \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x}\]

  5. Taylor expanded around inf 15.2

    \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \left(\cos x + 1\right)}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x + 1}}\]

  7. Final simplification0.3

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))